1. Modeling and Simulation
Dr. X

2. Topics
Random Numbers
Monte Carlo Simulation

3. Monte Carlo Simulation
Probabilistic model
Element of chance
Rare event simulation

4. John von Neumann Quantum theory Logic Game theory Economics
Theory and applications of Computer Science
Monte Carlo Simulation
WWII
How system neutrons would behave

5. Multiplicative linear congruential generator
Lehmer
Generate 10 random numbers
r0=10
rn = (7rn-1+1)mod11, n>0
What is r0?
How do we know it will only generate 10 numbers?

6. Ranges of random numbers
Math.random: 0<r<1
What if we want a range 0…99? 100…500?

7. Throwing darts Calculus: calculate area
What would you do if area cannot be calculated?
𝑓 𝑥 = cos⁡(𝑥) 2 +1

8. Throwing darts 𝑎𝑟𝑒𝑎=(𝑎𝑟𝑒𝑎 𝑜𝑓 𝑒𝑛𝑐𝑙𝑜𝑠𝑖𝑛𝑔 𝑟𝑒𝑐𝑡𝑎𝑛𝑔𝑙𝑒) #𝑑𝑎𝑟𝑡𝑠 𝑏𝑒𝑙𝑜𝑤 # 𝑑𝑎𝑟𝑡𝑠
𝑎𝑟𝑒𝑎=(𝑎𝑟𝑒𝑎 𝑜𝑓 𝑒𝑛𝑐𝑙𝑜𝑠𝑖𝑛𝑔 𝑟𝑒𝑐𝑡𝑎𝑛𝑔𝑙𝑒) #𝑑𝑎𝑟𝑡𝑠 𝑏𝑒𝑙𝑜𝑤 # 𝑑𝑎𝑟𝑡𝑠
The more darts we throw, the more accurate the results

9. Monte Carlo Algorithm
Monte Carlo algorithm to estimate area between x-axis and f from SMALLEST_X to LARGEST_X
MAX_Y  number greater than or equal to all f(x) in the interval
NUMDARTS  number of darts to be thrown
NumHitsBelow  0
Do the following NUMDARTS times:
randomX  random number between SMALLEST_X and LARGEST_X
randomY  random number between 0 and MAX_Y
if(randomY < f(randomX))
increment NumHitsBelow by 1
AreaOfRectangle  (LARGEST_X – SMALLEST_X) *(MAX_Y)
EstimatedArea  (AreaOfRectangle)*(NumHitsBelow)/ NUMDARTS

10. Random Numbers from Distributions
Distribution: description of the portion of times each possible outcome or each possible range of outcomes occurs on average
Discrete
Continuous

11. Generate random numbers from discrete distribution
Generate rand, a uniform random floating point number [0, 1)
If rand < p1, then return e1
else if rand < p1+ p2, then return e2 …
else if rand < p1+ p2+…+pn-1, then return en-1
else return en

12. Generate random numbers from discrete distribution

13. Generate random numbers from continuous distribution
Reverse the distribution
Normal – Box Muller
x = b * sin(a) + μ and x = b * cos(a) + μ where:
a = random number between [0, 2π)
rand = uniform random number between [0,1)
b = σ ln⁡(𝑟𝑎𝑛𝑑)
Exponential
x = 1/a * log(rand) where:

14. References
Introduction to computational science, Angela Shiflet, George Shiflet, Princeton University Press